Strehler-Mildvan Correlation Research Articles

Estimating network changes from lifespan measurements using a parsimonious gene network model of cellular aging

BackgroundCellular aging is best studied in the budding yeast Saccharomyces cerevisiae. As an example of a pleiotropic trait, yeast lifespan is influenced by hundreds of interconnected genes. However, no quantitative methods are currently available to infer system-level changes in gene networks during cellular aging.ResultsWe propose a parsimonious mathematical model of cellular aging based on stochastic gene interaction networks. This network model is made of only non-aging components: the strength of gene interactions declines with a constant mortality rate. Death of a cell occurs in the model when an essential node loses all of its interactions with other nodes, and is equivalent to the deletion of an essential gene. Stochasticity of gene interactions is modeled using a binomial distribution. We show that the exponential increase of mortality rate over time can emerge from this gene network model during the early stages of aging.We developed a maximal likelihood approach to estimate three lifespan-influencing network parameters from experimental lifespans: t0, the initial virtual age of the network system; n, the average lifespan-influencing interactions per essential node; and R, the initial mortality rate. We applied this model to yeast mutants with known effects on replicative lifespans. We found that deletion of SIR2, FOB1, and HXK2 considerably altered the initial virtual age but not the average lifespan-influencing interactions per essential node, suggesting that these mutations mainly influence the reliability of gene interactions but not the overall configurations of gene networks.We applied this model to investigate replicative lifespans of yeast natural isolates. We estimated that the average number of lifespan-influencing interactions per essential node is 7.0 (6.1–8) and the average estimated initial virtual age is 45.4 (30.6–74) cell divisions in these isolates. We also found that t0 could potentially mediate the observed Strehler-Mildvan correlation in yeast natural isolates.ConclusionsOur theoretical model provides a parsimonious interpretation of experimental lifespan data from the perspective of gene networks. We hope that our work will stimulate more interest in developing network models to study aging as a pleiotropic trait.

A 2D analysis of correlations between the parameters of the Gompertz–Makeham model (or law?) of relationships between aging, mortality, and longevity

When mortality (μ), aging rate (γ) and age (t) are treated according to the Gompertz model μ(t) = μ0eγt (GM), any mean age corresponds to a manifold of paired reciprocally changing μ0 and γ. Therefore, any noisiness of data used to derive GM parameters makes them negatively correlated. Besides this artifactual factor of the Strehler-Mildvan correlation (SMC), other factors emerge when the age-independent mortality C modifies survival according to the Gompertz-Makeham model μ(t) = C+μ0eγt (GMM), or body resources are partitioned between survival and protection from aging [the compensation effect of mortality (CEM)]. Theoretical curves in (γ, logμ0) coordinates show how μ0 decreases when γ increases upon a constant mean age. Within a species-specific range of γ, such "isoage" curves look as nearly parallel straight lines. The slopes of lines constructed by applying GM to survival curves modeled according to GMM upon changes in C are greater than the isoage slopes. When CEM is modeled, the slopes are still greater. Based on these observations, CEM is shown to contribute to SMC associated with sex differences in lifespan, with the effects of several life-extending drugs, and with recent trends in survival/mortality patterns in high-life-expectancy countries; whereas changes in C underlie differences between even high-life-expectancy countries, not only between high- and low-life-expectancy countries. Such interpretations make sense only if GM is not merely a statistical model, but rather reflects biological realities. Therefore, GM is discussed as derivable by applying certain constraints to a natural law termed the generalized Gompertz-Makeham law.

Gender Specificity of the Effect of Neonatal Melatonin Administration on Lifespan and Age-Associated Pathology in 129/Sv Mice

Melatonin is subcutaneously administered in a single dose of 1.2 μg to 129/Sv mice at days 3, 5, and 7 after birth, and the mice are observed until natural death. In adult males, a decrease in the body weight and a reduction of the contribution of lung lesions detected during postmortal autopsy to mortality are detected. In adult females, an increase in the proportion of mice with impaired estrous cycle at the later stages of life and an acceleration of the mortality associated with uterine hemangioma are observed with the absence of any effect of melatonin on the body weight. Changes in the frequency of detection of other tumors are multidirectional: there is a decrease in the frequency in males and an increase in females. The age of death of the last 25% and 10% of males increases, while in females it decreases. An analysis of the complete survivorship curves using the Gompertz model does not reveal changes in the rate of aging and initial mortality sufficient to go beyond the limits determined by the artifact component of the Strehler–Mildvan correlation between these parameters. In general, the tendencies noted in males and females are multidirectional: favorable and unfavorable, respectively. Melatonin is contraindicated for pregnant and nursing women and children under 18 years old. The gender specificity of the effects of melatonin, apparently, should be considered when assessing the feasibility of its prescription to these categories.

In mice transgenic for IGF1 under keratin-14 promoter, lifespan is decreased and the rates of aging and thymus involution are accelerated.

IGF1 signaling is supposedly a key lifespan determinant in metazoans. However, controversial lifespan data were obtained with different means used to modify IGF1 or its receptor (IGF1R) expression in mice. The emerging puzzle lacks pieces of evidence needed to construct a coherent picture. We add to the available evidence by using the Gompertz model (GM), with account for the artifactual component of the Strehler-Mildvan correlation between its parameters, to compare the survival patterns of female FVB/N and FVB/N-derived K14/mIGF1 mice. In K14/mIGF1 vs. FVB/N mice, the rate of aging (γ) is markedly increased without concomitant changes in the initial mortality (μ0). In published cases where IGF1 signaling was altered by modifying liver or muscle IGF1 or whole body IGF1R expression, lifespan changes are attributable to μ0. The accelerated aging and associated tumor yield in K14/mIGF1 mice are consistent with the finding that the age-associated decreases in thymus weight and serum thymulin are accelerated in K14/mIGF1 mice. Our results underscore the importance of accounting for the mathematical artifacts of data fitting to GM in attempts to resolve discrepancies in survival data and to differentiate the contributions of the initial mortality and the rate of aging to changes in lifespan.

Applying parametric models to survival data: tradeoffs between statistical significance, biological plausibility, and common sense.

Parametric models for survival data help to differentiate aging from other lifespan determinants. However, such inferences suffer from small sizes of experimental animal samples and variable animals handling by different labs. We analyzed control data from a single laboratory where interventions in murine lifespan were studied over decades. The minimal Gompertz model (GM) was found to perform best with most murine strains. However, when several control datasets related to a particular strain are fitted to GM, strikingly rigid interdependencies between GM parameters emerge, consistent with the Strehler-Mildvan correlation (SMC). SMC emerges even when survival patterns do not conform to GM, as with cancer-prone HER2/neu mice, which die at a log-normally distributed age. Numerical experiments show that SMC includes an artifact whose magnitude depends on dataset deviation from conformance to GM irrespectively of the noisiness of small datasets, another contributor to SMC. Still another contributor to SMC is the compensation effect of mortality (CEM): a real tradeoff between the physiological factors responsible for initial vitality and the rate of its decline. To avoid misinterpretations, we advise checking experimental results against a SMC based on historical controls or on subgroups obtained by randomization of control animals. An apparent acceleration of aging associated with a decrease in the initial mortality is invalid if it is not greater than SMC suggests. This approach applied to published data suggests that the effects of calorie restriction and of drugs believed to mimic it are different. SMC and CEM relevance to human survival patterns is discussed.

Strehler-Mildvan correlation is a degenerate manifold of Gompertz fit

Gompertz empirical law of mortality is often used in practical research to parametrize survival fraction as a function of age with the help of just two quantities: the Initial Mortality Rate (IMR) and the Gompertz exponent, inversely proportional to the Mortality Rate Doubling Time (MRDT). The IMR is often found to be inversely related to the Gompertz exponent, which is the dependence commonly referred to as Strehler-Mildvan (SM) correlation. In this paper, we address fundamental uncertainties of the Gompertz parameters inference from experimental Kaplan-Meier plots and show, that a least squares fit often leads to an ill-defined non-linear optimization problem, which is extremely sensitive to sampling errors and the smallest systematic demographic variations. Therefore, an analysis of consequent repeats of the same experiments in the same biological conditions yields the whole degenerate manifold of possible Gompertz parameters. We find that whenever the average lifespan of species greatly exceeds MRDT, small random variations in the survival records produce large deviations in the identified Gompertz parameters along the line, corresponding to the set of all possible IMR and MRDT values, roughly compatible with the properly determined value of average lifespan in experiment. The best fit parameters in this case turn out to be related by a form of SM correlation. Therefore, we have to conclude that the combined property, such as the average lifespan in the group, rather than IMR and MRDT values separately, may often only be reliably determined via experiments, even in a perfectly homogeneous animal cohort due to its finite size and/or low age-sampling frequency, typical for modern high-throughput settings. We support our findings with careful analysis of experimental survival records obtained in cohorts of C. elegans of different sizes, in control groups and under the influence of experimental therapies or environmental conditions. We argue that since, SM correlation may show up as a consequence of the fitting degeneracy, its appearance is not limited to homogeneous cohorts. In fact, the problem persists even beyond the simple Gompertz mortality law. We show that the same degeneracy occurs exactly in the same way, if a more advanced Gompertz-Makeham aging model is employed to improve the modeling. We explain how SM type of relation between the demographic parameters may still be observed even in extremely large cohorts with immense statistical power, such as in human census datasets, provided that systematic historical changes are weak in nature and lead to a gradual change in the mean lifespan.

Evolutionary theory of ageing and the problem of correlated Gompertz parameters

The Gompertz mortality model is often used to evaluate evolutionary theories of ageing, such as the Medawar–Williams' hypothesis that high extrinsic mortality leads to faster ageing. However, fits of the Gompertz mortality model to data often find the opposite result that mortality is negatively correlated with the rate of ageing. This negative correlation has been independently discovered in several taxa and is known in actuarial studies of ageing as the Strehler–Mildvan correlation. We examine the role of mortality selection in determining late-life variation in susceptibility to death, which has been suggested to be the cause of this negative correlation. We demonstrate that fixed-frailty models that account for heterogeneity in frailty do not remove the correlation and that the correlation is an inherent statistical property of the Gompertz distribution. Linking actuarial and biological rates of ageing will continue to be a pressing challenge, but the Strehler–Mildvan correlation itself should not be used to diagnose any biological, physiological, or evolutionary process. These findings resolve some key tensions between theory and data that affect evolutionary and biological studies of ageing and mortality. Tests of evolutionary theories of ageing should include direct measures of physiological performance or condition.

An informative prior probability distribution of the gompertz parameters for bayesian approaches in paleodemography.

In paleodemography, the Bayesian approach has been suggested to provide an effective means by which mortality profiles of past populations can be adequately estimated, and thus avoid problems of "age-mimicry" inherent in conventional approaches. In this study, we propose an application of the Gompertz model using an "informative" prior probability distribution by revising a recent example of the Bayesian approach based on an "uninformative" distribution. Life-table data of 134 human populations including those of contemporary hunter-gatherers were used to determine the Gompertz parameters of each population. In each population, we used both raw life-table data and the Gompertz parameters to calculate some demographic values such as the mean life-span, to confirm representativeness of the model. Then, the correlation between the two Gompertz parameters (the Strehler-Mildvan correlation) was re-established. We incorporated the correlation into the Bayesian approach as an "informative" prior probability distribution, and tested its effectiveness using simulated data. Our analyses showed that the mean life-span (≥ age 15) and the proportion of living persons aging over 45 were well-reproduced by the Gompertz model. The simulation showed that using the correlation as an informative prior provides a narrower estimation range in the Bayesian approach than does the uninformative prior. The Gompertz model can be assumed to accurately estimate the mean life-span and/or the proportion of old people in a population. We suggest that the Strehler-Mildvan correlation can be used as a useful constraint in demographic reconstructions of past human populations.

Life equations for the senescence process

The Gompertz law of mortality quantitatively describes the mortality rate of humans and almost all multicellular animals. However, its underlying kinetic mechanism is unclear. The Gompertz law cannot explain the mortality plateau at advanced ages and cannot give an explicit relationship between temperature and mortality. In this study a reaction kinetics model with a time dependent rate coefficient is proposed to describe the survival and senescence processes. A temperature-dependent mortality function was derived. The new mortality function becomes the Gompertz mortality function with the same relationship of parameters prescribed by the Strehler–Mildvan correlation when age is smaller than a characteristic value δ, and reaches the mortality plateau when age is greater than δ. A closed-form analytical expression for describing the relationship of average lifespan with temperature and other equations are derived from the new mortality function. The derived equations can be used to estimate the limit of average lifespan, predict the maximal longevity, calculate the temperature coefficient of lifespan, and explain the tendency of the survival curve. This prediction is consistent with the most recently reported mortality trajectories for single-year birth cohorts. This study suggests that the senescence process results from the imbalance between damaging energy and protecting energy for the critical chemical substance in the body. The rate of senescence of the organism increases while the protecting energy decreases. The mortality plateau is reached when the protecting energy decreases to its minimal levels. The decreasing rate of the protecting energy is temperature-dependent. This study is exploring the connection between the biochemical mechanism and demography.

The Strehler–Mildvan correlation from the perspective of a two-process vitality model

The Strehler and Mildvan (SM) general theory of ageing and mortality provides a mechanism-based explanation of Gompertz's law and predicts a log-linear relationship between the two Gompertz coefficients, known as the SM correlation. While the SM correlation is supported by data from developed countries before the second half of the twentieth century, the recent breakdown of the correlation pattern in these countries has prompted demographers to conclude that SM theory needs to be reassessed. In this paper we use a newly developed two-process vitality model to explain the SM correlation and its breakdown in terms of asynchronous trends in acute (extrinsic) and chronic (intrinsic) mortality factors. We propose that the mortality change in the first half of the twentieth century is largely determined by the elimination of immediate hazards to death, whereas the mortality change in the second half is primarily driven by the slowdown of the deterioration rate of intrinsic survival capacity.

The impact of environment on lifespan distribution dynamics

The environment generates numerous signals that control the lifespan of species living in it, including humans. By approximating survival curves with the Gompertz function and judging by the Strehler-Mildvan correlation (a linear dependence between the Gompertz function parameters), we have constructed 3D surfaces of survival curves in a wide range of controlling parameter values: characteristic species lifespan and cohort death rate. It has turned out that these surfaces have a complex form. In particular, we have shown the nonlinearity of the dependence between such human lifespan distribution parameters as mean lifespan and variance.

A New Model of Ageing and Mortality

ABSTRACTThe Nested Binomial Model presented in this paper is a new approach to modelling mortality and survival in humans and other species that seeks to reconcile individual life course risk trajectories and those population mortality patterns that arise from inter-individual heterogeneity. In describing individual trajectories it partitions mortality risk into two main elements: ‘redundancy’ and ‘interactive risk'. Interactive risk is volatile, increasing or decreasing with time and circumstance, while redundancy is a quantity which declines in a linear and largely invariable fashion throughout life. Although a biological correlate for redundancy is not identified, this assumption allows strikingly realistic modelling of mortality and survival curves, late-life mortality deceleration, Strehler-Mildvan correlation, mortality plateaux and slowing of mortality. Simple assumptions with regard to heterogeneity of parameters within the model allow close approximation to the entire human mortality curve, and provide a rationale for observed and otherwise paradoxical population mortality phenomena. As such, it fulfils biodemographic criteria for a comprehensive theory of ageing. Future challenges are to reconcile its theoretical structure with empirical findings in the biology of ageing and to render it in a form that can become a usable actuarial tool.

The natural distribution of survival

Great similarities in survival patterns permit the Gompertz and other established equations to describe parts of mortality curves in various species. These patterns appear non-random and invite inference of biological meaning, though no unifying explanation is agreed. Under the theory described here, linear decline of an initial quantity of species or strain-specific redundant reserve interacts with extrinsic hazards via a ‘nested binomial’ model, which is presented both in a simple, four parameter form, and a more complex form that incorporates inter-individual and inter-functional biological variation. This approach demonstrates exponential rises in mortality, late-life deceleration and Strehler–Mildvan correlation. Biological variation within the complex model, specifically in the redundancy decay rate parameter, is shown to generate mortality plateaux, while outlier phenotypes produce mortality decelerations, supporting inter-individual heterogeneity as the cause of these phenomena. The model is robust to large variations in organism complexity, and to broad intra-population hazard variation. Specific parameters appear analogous to observed elements of ageing, and a central role for redundancy depletion provides a context for longevity genes and rapid evolution of increased lifespan. This approach offers a unifying model for a great variety of ageing phenomena across a wide range of species.

Does Makeham make sense?

Numerical modeling was used to explore the behavior of ideal cohorts obeying the Gompertz-Makeham (GM) law of mortality (-dn/dt 1/n(t) = C + lambda egammat) supplemented with the Strehler-Mildvan (SM) correlation (ln lambda = A - Bgamma) and to show how changes in the age-independent parameter C will produce an apparent SM correlation if C is ignored in mortality data treatment as in the case of the so-called longitudinal gompertzian analysis of historical changes in human mortality patterns. The essential difference between the Makeham term C and Gompertz term lambda e(gammat) has been suggested to be not that the latter is age-dependent whereas the former is not, but that C comprises the contributions of inherently irresistible stresses to mortality, whereas lambda e(gammat) comprises the contributions of resistible stresses and shows how changes in the resistance to them are translated into changes in mortality. This assumption was used to show by modeling how the transition of stresses from irresistible to resistible may result in decreased late survivorship as the cost of increased early survivorship, in line with the antagonistic pleiotropy theory of aging. On the whole, the modeling suggests that the GM equation is not only a mathematical tool for treatment of mortality data but that it also has a fundamental biological significance, and its Makeham term C should not be ignored in any analysis of mortality data.

Do intersections of mortality-rate and survival functions have significance?

Common points of intersections have frequently been reported among members of families of linearized mortality-rate and survival functions. A general condition for the existence of such intersections is derived. It is shown that a common point of intersection between straight-line functions exists if and only if the intercepts of the functions are linearly related to their slopes. This slope-intercept condition is applied to a didactic model to illustrate its generality and to three models, the Gompertz-Makeham, the Weibull, and the logistic, which are often used in the analysis of mortality data. The slope-intercept condition for the Gompertz-Makeham mortality-rate model proves to be the well-known Strehler-Mildvan correlation. Families of mortality-rate functions or of the corresponding survival functions but not both may display common points of intersection. Differences between the ages at which survival functions intersect and those at which the associated mortality-rate functions intersect are calculated to be of the order of magnitude of 10 to 20 years. Survival function intersections lie close to the limit of human life span but often arise in consequence of unsupported extrapolations of data obtained at younger ages. These and other results lead to the conclusion that, in themselves, the intersections of survival and mortality-rate functions are not of great importance. To the extent that significance can be attributed to the intersections, it lies in the existence of linear relationships between their slopes and intercepts.